MathBench > Population Dynamics

Bacterial Growth:
The E. coli ate my homework

In Search of ... a more convenient equation

So now you know how to look at the data on a culture and find the doubling time. What about starting with the doubling time and predicting how the culture should grow? For example, if you know that the doubling time is 22 minutes and the culture grows for 66 minutes, its pretty obvious that it doubles 3 times. For example, if you started with an N0 of 2 million cells/ml, you would end with (approximately) 16 million cells/ml.

But can you predict the population size after 55 minutes? Well, yes, you could write:

Nt = N0 * 2g, so

Nt = 2 million * 2(2.5 generations) [since 55 minutes = 22*2.5, or 2.5 doubling times]

If we then want to compare that to something that doubles in 25 minutes, we would write

Nt = 2 million * 2(2.2 generations) [since 55 = 25*2.2]

What would be really nice would be to have population size as a function of minutes, rather than as a function of the number of generations that have gone by. Then we could simply write

Nt = (some function)55

for each population. But how to do this?

And while we're at it, mathematicians generally prefer to state growth equations with a base of "e", rather than a base of 2. Why? What's so special about e?

First of all, e is just a number -- actually it's a bit less than 3, a decimal that goes on forever without repeating. But its a pretty special number -- you know it has to be, seeing as it has a name. You probably know another named number, pi, which is very useful for calculating quantities related to circles, such as their circumference or area. Like e, pi is simply a number (in this case a bit more than 3), but its so useful that mathematicians gave it a name.

Pi is useful for circles, whereas e is useful for exponential equations. The reason is that when you graph the function

y = ex,

you get a curve in which the y value and the slope are always equal. Or, you could say, the population size (y) and the number added per unit time are the same. For example, if the population is 1000, then it is also adding 1000 individuals per unit of time (at least for a brief moment -- as soon as some individuals get added, the rate of adding also goes up).

Obviously this situation, where the y value and the slope are the same, is very unusual -- you need a very special base in the exponential equation to make it happen. That base is "e". Here's one way to determine the exact value of "e" graphically:

 

"e" is a great invention if you're a mathematician, because it makes the curve growth easier to analyze. For everyone else, we use "e" because mathematicians use "e".

Of course, just like very few circles have a diameter of exactly 1, very few populations follow the ex curve exactly. Luckily, with a few modifications, we can make this equation fit our needs.